(3 pontos) Se w=x^{4} y+y^{2} z^{3}, onde x=r s e^{t}, y=r s^{2} e^{-t} e z=r^{2} s \operatorname{sen}(t), determine o valor de \frac{\partial w}{\partial s} quando r=2, s=1 e t=0.
8. Dados a função f(x, y, z)=x e^{2 y z}, o ponto P(3,0,2) e o vetor u=\frac{2}{3} i-\frac{2}{3} j+\frac{1}{3} k.

Passo 1

O enunciado define
w = x^{4}y + y^{2}z^{3},
com
x = r\,s\,e^{t},\quad y = r\,s^{2}\,e^{-t},\quad z = r^{2}\,s\,\operatorname{sen}(t),
e pede \displaystyle \frac{\partial w}{\partial s} em r=2,\;s=1,\;t=0.
Conteúdo: derivada parcial de função composta (regra da cadeia).

Passo 2

Usamos a regra da cadeia para funções de várias variáveis:
\frac{\partial w}{\partial s} = \frac{\partial w}{\partial x}\,\frac{\partial x}{\partial s} + \frac{\partial w}{\partial y}\,\frac{\partial y}{\partial s} + \frac{\partial w}{\partial z}\,\frac{\partial z}{\partial s}.

Passo 3

Calculamos \partial w/\partial x, \partial w/\partial y e \partial w/\partial z a partir de
w = x^{4}y + y^{2}z^{3}:
\frac{\partial w}{\partial x} = 4x^{3}y,\quad \frac{\partial w}{\partial y} = x^{4} + 2y\,z^{3},\quad \frac{\partial w}{\partial z} = 3y^{2}z^{2}.

Passo 4

Agora \partial x/\partial s, \partial y/\partial s, \partial z/\partial s:
x = r\,s\,e^{t}\;\Rightarrow\;\frac{\partial x}{\partial s} = r\,e^{t},
y = r\,s^{2}\,e^{-t}\;\Rightarrow\;\frac{\partial y}{\partial s} = 2r\,s\,e^{-t},
z = r^{2}\,s\,\operatorname{sen}(t)\;\Rightarrow\;\frac{\partial z}{\partial s} = r^{2}\,\operatorname{sen}(t).

Passo 5

Avaliaremos tudo em r=2, s=1, t=0. Primeiro os valores de x,y,z:
x(2,1,0) = 2\cdot1\cdot e^{0} = 2,\quad y(2,1,0) = 2\cdot1^{2}\cdot e^{0} = 2,\quad z(2,1,0) = 2^{2}\cdot1\cdot\operatorname{sen}(0) = 0.
E as derivadas parciais de x,y,z em relação a s:
\frac{\partial x}{\partial s}\Big|_{(2,1,0)} = 2\,e^{0} = 2, \quad \frac{\partial y}{\partial s}\Big|_{(2,1,0)} = 2\cdot2\cdot1\cdot e^{0} = 4, \quad \frac{\partial z}{\partial s}\Big|_{(2,1,0)} = 2^{2}\,\operatorname{sen}(0) = 0.

Passo 6

Substituindo tudo na regra da cadeia:
\frac{\partial w}{\partial s} = \Bigl(4x^{3}y\Bigr)\cdot2 + \Bigl(x^{4}+2y\,z^{3}\Bigr)\cdot4 + \Bigl(3y^{2}z^{2}\Bigr)\cdot0.
Em (x,y,z)=(2,2,0) e z=0 simplifica:
4x^{3}y = 4\cdot2^{3}\cdot2 = 4\cdot8\cdot2 = 64, \quad x^{4}+2y\,z^{3} = 2^{4}+2\cdot2\cdot0 = 16.
Então
\frac{\partial w}{\partial s} = 64\cdot2 + 16\cdot4 = 128 + 64 = 192.

Resposta

\displaystyle 192

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